Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization (1606.04933v1)

Abstract: We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\ast g$. This problem, known as {\em blind deconvolution}, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on $f$ and $g$. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.

• The paper introduces a computationally efficient algorithm using regularized gradient descent within a nonconvex optimization framework to solve the blind deconvolution problem.
• The proposed algorithm provides rigorous theoretical guarantees for exact signal recovery under specific conditions and demonstrates robustness in the presence of noise.
• This nonconvex method offers computational superiority over existing convex techniques, making it a rapid and reliable solution for applications in imaging and communications.

Overview of "Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization"

The paper "Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization" explores the challenge of blind deconvolution, which involves reconstructing two signals from their convolution when both signals are unknown. This problem is prevalent in various scientific and engineering fields, such as astronomy, medical imaging, and communications. The primary difficulty lies in the non-convex nature of the problem, which is prone to numerous local minima, posing significant challenges for conventional optimization techniques.

The authors introduce a computationally efficient algorithm based on regularized gradient descent, which leverages nonconvex optimization to address these challenges. They demonstrate that their algorithm is capable of exact recovery under near-optimal measurement conditions, exhibiting robustness in the presence of noise. The paper claims that the proposed method is the first of its kind to provide both numerical efficiency and rigorous recovery guarantees within reasonably defined subspaces of the signals.

Key Contributions

1. Algorithm Development: The authors propose a regularized gradient descent algorithm that efficiently converges to the true solution of the blind deconvolution problem. The algorithm is designed to work under practical conditions where the number of measurements slightly exceeds the information-theoretical minimum required for recovery.
2. Theoretical Guarantees: The regularized gradient descent approach comes with mathematical proofs of its convergence, ensuring that the method does not get trapped in local minima. The authors provide explicit conditions under which the algorithm is guaranteed to succeed, solidifying its reliability.
3. Subspace Assumptions: To handle the inherently ill-posed nature of blind deconvolution, the paper assumes that the signals belong to known subspaces. This assumption allows for a robust and flexible mathematical framework amenable to rigorous analysis.
4. Empirical Validation: Through numerical experiments, the paper not only confirms the theoretical findings but also showcases the algorithm’s performance in scenarios beyond those covered by the theoretical framework. The empirical results highlight the robustness and effectiveness of the proposed solution even with noisy data, underscoring its practical applicability.
5. Comparison with Existing Methods: The paper positions the proposed approach as superior to existing convex optimization methods, often limited by high computational costs. The simulations suggest that the nonconvex method requires fewer measurements than convex approaches, providing a significant advantage in terms of efficiency.

Implications and Future Directions

The implications of this research are far-reaching in the field of signal processing and related areas. By providing a reliable and fast alternative to blind deconvolution challenges, this work could enhance applications that depend heavily on signal reconstruction, such as better imaging techniques in medical diagnostics and more efficient communication systems.

Looking forward, this work opens several avenues for further exploration. One potential area is the extension of these techniques to other forms of signal and image processing problems, such as blind calibration and demixing. Additionally, exploring the theoretical guarantees for broader classes of subspace structures or alternative problem settings could expand the applicability of these techniques.

In conclusion, this work presents a significant step forward in addressing blind deconvolution problems by introducing a new method that combines computational efficiency with strong theoretical foundations. The proposed framework not only adds to the toolkit of signal processing techniques but also encourages further research into nonconvex optimization methods for complex inverse problems.